Smearing Retransformation
Can’t use normal theory restransformation from logs to dollars \((\exp(\sigma^2/2))\) because the error distributions of the OLS log regressions deviate from the normal assumption.
Nonparametric estimate of the retransformation of factors (Duan, 1983),
\[
\hat{\phi_j}=\Sigma_i \exp(\hat{\varepsilon_{ij}})/n_j, \; j=3,4,
\]
where \(n_j\) is sample size of equation \(j\), \(\hat{\varepsilon}_{ij}=\ln(y_{ij})-x_{ij}\hat{\beta}_j\), and \(\hat{\beta_j}\) is the OLS estimate, \(i\) indexes the person.